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Optimization in Medicine Springer Series in Optimization and Its Applications VOLUME12 ManagingEditor PanosM.Pardalos(UniversityofFlorida) Editor—CombinatorialOptimization Ding-ZhuDu(UniversityofTexasatDallas) AdvisoryBoard J.Birge(UniversityofChicago) C.A.Floudas(PrincetonUniversity) F.Giannessi(UniversityofPisa) H.D.Sherali(VirginiaPolytechnicandStateUniversity) T.Terlaky(McMasterUniversity) Y.Ye(StanfordUniversity) AimsandScope Optimization has been expanding in all directions at an astonishing rate during the lastfewdecades.Newalgorithmicandtheoreticaltechniqueshavebeendeveloped, thediffusionintootherdisciplineshasproceededatarapidpace,andourknowledge ofallaspectsofthefieldhasgrownevenmoreprofound.Atthesametime,oneof themoststrikingtrendsinoptimizationistheconstantlyincreasingemphasisonthe interdisciplinarynatureofthefield.Optimizationhasbeenabasictoolinallareasof appliedmathematics,engineering,medicine,economicsandothersciences. TheSpringerSeriesinOptimizationandItsApplicationspublishesundergradu- ate and graduate textbooks, monographs and state-of-the-art expository works that focus on algorithms for solving optimization problems and also study applications involvingsuchproblems.Someofthetopicscoveredincludenonlinearoptimization (convex and nonconvex), network flow problems, stochastic optimization, optimal control,discreteoptimization,multi-objectiveprogramming,descriptionofsoftware packages,approximationtechniquesandheuristicapproaches. Carlos J. S. Alves Panos M. Pardalos Luis Nunes Vicente Editors Optimization in Medicine ABC Editors CarlosJ.S.Alves PanosM.Pardalos InstitutoSuperiorTécnico DepartmentofIndustrialandSystems Av.RoviscoPais1 Engineering 1049-001Lisboa UniversityofFlorida Portugal 303WeilHall Gainesville,FL32611 USA LuisNunesVicente DepartamentodeMatemática FaculdadedeCiênciaseTecnologia UniversidadedeCoimbra 3001-454Coimbra Portugal ManagingEditor: Editor/CombinatorialOptimization PanosM.Pardalos Ding-ZhuDu UniversityofFlorida UniversityofTexasatDallas ISBN978-0-387-73298-5 e-ISBN978-0-387-73299-2 DOI:10.1007/978-0-387-73299-2 LibraryofCongressControlNumber:2007934793 MathematicsSubjectClassification(2000):49XX,46N60 (cid:1)c 2008SpringerScience+BusinessMedia,LLC Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permissionofthepublisher(SpringerScience+BusinessMedia,LLC,233SpringStreet,NewYork,NY 10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Useinconnection withanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject toproprietaryrights. Printedonacid-freepaper. 9 8 7 6 5 4 3 2 1 springer.com Preface Optimizationhasbecomepervasiveinmedicine.Theapplicationofcomputing to medical applications has opened many challengingissues and problems for both the medical computing field and the mathematical community. Mathe- matical techniques (continuous and discrete) are playing a key role with increasing importance in understanding several fundamental problems in medicine.Naturally,optimizationisafundamentallyimportanttoolduetothe limitation of the resources involved and the need for better decision making. The book startswith twopapers onIntensity Modulated RadiationTher- apy(IMRT). Thefirstpaper,byR.Acosta,M.Ehrgott,A.Holder,D.Nevin, J. Reese, and B. Salter, discusses an important subproblem in the design of radiation plans, the selection of beam directions. The manuscript compares different heuristic methods for beam selection on a clinical case and stud- ies the effect of various dose calculation grid resolutions. The next paper, by M.Ehrgott,H.W.Hamacher,andM.Nußbaum,reviewsseveralcontributions onthedecompositionofmatricesasamodelforrearrangingleavesonamulti- leafcollimator.SuchaprocessisessentialforblockradiationinIMRTinorder to achieve desirable intensity profiles. Additionally, they present a new ap- proachfor minimizing the number of decompositionsegmentsby sequentially solving this problem in polynomial time with respect to fixed decomposition times. The book continues with a paper by G. Deng and M. Ferris on the formulation of the day-to-day radiation therapy treatment planning problem as a dynamic program. The authors consider errors due to variations in the positioningofthepatientandapplyneuro-dynamicprogrammingtocompute approximate solutions for the dynamic optimization problems. The fourth paper,by H.Fohlin,L.Kliemann,andA.Srivastav,considersthe seedrecon- struction problem in brachytherapy as a minimum-weight perfect matching probleminahypergraph.Theproblemismodeledasanintegerlinearprogram for which the authors develop an algorithm based on a randomized rounding scheme and a greedy approach. VI Preface The book also covers other types of medical applications. For instance, in thepaperbyS.Sabesan,N.Chakravarthy,L.Good,K.Tsakalis,P.Pardalos, and L. Iasemidis, the authors propose an application of global optimization in the selection of critical brain sites prior to an epileptic seizure. The paper shows the advantages of using optimization (in particular nonconvex quadratic programming) in combination with measures of EEG dynamics, such as Lyapunov exponents, phase and energy, for long-term prediction of epileptic seizures. E. K. Lee presents the optimization-classification models within discrim- inant analysis, to develop predictive rules for large heterogeneous biological and medical data sets. As mentioned by the author, classification models are critical to medical advances as they can be used in genomic, cell molecular, and system level analysis to assist in early prediction, diagnosis and detec- tion of diseases, as well as for intervention and monitoring. A wide range of applications are described in the paper. This book also includes two papers on inverse problems with applications tomedicalimaging.ThepaperbyA.K.Louispresentsanoverviewofseveral techniquesthatleadtorobustalgorithmsforimagingreconstructionfromthe measured data. In particular, the inversion of the Radon transform is con- sidered as a model case of inversion. In this paper, a reconstruction of the inside of a surprise egg is presented as a numerical example for 3D X-Ray reconstructionfromrealdata.Inthe paper byM.Malinen,T.Huttunen, and J.Kaipio,aninverseproblemrelatedtoultrasoundsurgeryisconsideredinan optimization frameworkthat aims to controlthe optimal thermal dose to ap- ply,forinstance,inthetreatmentofbreastcancer.Twoalternativeprocedures (a scanning path optimization algorithm and a feedforward-feedback control method) are discussed in detail with numerical examples in 2D and 3D. We would like to thank the authors for their contributions. It would not have been possible to reach the quality of this publication without the contributions of the many anonymous referees involved in the revision and acceptance process of the submitted manuscripts. Our gratitude is extended to them as well. This book was generated mostly from invited talks given at the Work- shop on Optimization in Medicine, July 20-22,2005, which took place at the Institute of Biomedical Research in Light and Image (IBILI), University of Coimbra, Portugal. The workshop was organized under the auspices of the International Center for Mathematics (CIM, http://www.cim.pt)as part of the 2005 CIM Thematic Term on Optimization. Finally, we would like to thank Ana Lu´ısa Custódio (FCT/UNL) for her helpintheorganizationoftheworkshopandPedroC.Martins(ISCAC/IPC) and Joaõ M. M. Patr´ıcio (ESTT/IPT) for their invaluable editorial support. Coimbra, C. J. S. Alves May 2007 P. M. Pardalos L. N. Vicente Contents The influence of dose grid resolution on beam selection strategies in radiotherapy treatment design Ryan Acosta, Matthias Ehrgott, Allen Holder, Daniel Nevin, Josh Reese, and Bill Salter............................................. 1 Decomposition of matrices and static multileaf collimators: a survey Matthias Ehrgott, Horst W. Hamacher, and Marc Nußbaum ........... 25 Neuro-dynamic programming for fractionated radiotherapy planning Geng Deng and Michael C. Ferris.................................. 47 Randomized algorithms for mixed matching and covering in hypergraphs in 3D seed reconstruction in brachytherapy Helena Fohlin, Lasse Kliemann, and Anand Srivastav ................ 71 Global optimization and spatial synchronization changes prior to epileptic seizures Shivkumar Sabesan, Levi Good, Niranjan Chakravarthy, Kostas Tsakalis, Panos M. Pardalos, and Leon Iasemidis ....................103 Optimization-basedpredictive modelsin medicine and biology Eva K. Lee......................................................127 Optimal reconstruction kernels in medical imaging Alfred K. Louis ..................................................153 Optimal control in high intensity focused ultrasound surgery Tomi Huttunen, Jari P. Kaipio, and Matti Malinen ..................169 List of Contributors Ryan Acosta Helena Fohlin Institute for Computational and Department of Oncology, Linko¨ping Mathematical Engineering, University Hospital, Stanford University 581 85 Linko¨ping, Sweden. Stanford, California, USA. [email protected] [email protected] Levi Good Niranjan Chakravarthy The Harrington Department of Department of Electrical Engineer- Bioengineering, Fulton School of ing, Fulton School of Engineering, Engineering, Arizona State Arizona State University, University, Tempe, AZ 85281,USA. Tempe, AZ 85281,USA. [email protected] [email protected] Horst W. Hamacher Geng Deng Fachbereich Mathematik, Technische Department of Mathematics, Universita¨t Kaiserslautern, University of Wisconsin at Madison, Kaiserslautern,Germany. 480Lincoln Dr., Madison,WI 53706, [email protected] USA. [email protected] Allen Holder Matthias Ehrgott Department of Mathematics, Trinity Department of Engineering Science, University, and The University of Auckland, Department of Radiation Oncology, Auckland, New Zealand. University of Texas Health Science Center, San Antonio, Texas, [email protected] USA. Michael C. Ferris [email protected] Computer Sciences Department, University of Wisconsin at Madison, Tomi Huttunen 1210 W. Dayton Street, Department of Physics,University of Madison, WI 53706,USA. Kuopio, P.O. Box 1627, [email protected] FIN-70211,Finland. X List of Contributors Leon Iasemidis Marc Nußbaum The Harrington Department of Fachbereich Mathematik, Technische Bioengineering, Fulton School of Universita¨t Kaiserslautern, Engineering, Arizona State Kaiserslautern, Germany. University, Tempe, AZ 85281, USA. [email protected] Panos M. Pardalos Department of Industrial and Jari P. Kaipio Systems Engineering, Department of Physics, University of University of Florida, Gainesville, Kuopio, P.O. Box 1627, FL 32611, USA. FIN-70211, Finland. [email protected] Lasse Kliemann Josh Reese Institut fu¨r Informatik, Christian– Department of Mathematics, Albrechts–Universita¨t zu Kiel, Trinity University, Christian-Albrechts-Platz 4, San Antonio, Texas, USA. D–24098 Kiel, Germany. [email protected] [email protected] Shivkumar Sabesan Eva K. Lee Department of Electrical Engineer- Center for Operations Research in ing, Fulton School of Engineering, Medicine and HealthCare, Arizona State University, School of Industrial and Tempe, AZ 85281, USA. Systems Engineering, [email protected] Georgia Institute of Technology, Atlanta, GA 30332-0205, USA. Bill Salter [email protected] Department of Radiation Oncology, University of Utah Huntsman Alfred K. Louis Cancer Institute, Department of Mathematics, Salt Lake City, Utah, USA. Saarland University, [email protected] 66041 Saarbru¨cken, Germany. [email protected] Anand Srivastav Institut fu¨r Informatik, Christian– Matti Malinen Albrechts–Universita¨t zu Kiel, Department of Physics, University of Christian-Albrechts-Platz 4, Kuopio, P.O. Box 1627, D–24098 Kiel, Germany. FIN-70211, Finland. [email protected] [email protected] Kostas Tsakalis Daniel Nevin Department of Electrical Engineer- Department of Computer Science, ing, Fulton School of Engineering, Texas A&M University, Arizona State University, Tempe, College Station, Texas, USA. AZ 85281, USA. [email protected] [email protected]